Speech-based algorithms have gained interest for the management of behavioral health conditions such as depression. We explore a speech-based transfer learning approach that uses a lightweight encoder and that transfers only the encoder weights, enabling a simplified run-time model. Our study uses a large data set containing roughly two orders of magnitude more speakers and sessions than used in prior work. The large data set enables reliable estimation of improvement from transfer learning. Results for the prediction of PHQ-8 labels show up to 27% relative performance gains for binary classification; these gains are statistically significant with a p-value close to zero. Improvements were also found for regression. Additionally, the gain from transfer learning does not appear to require strong source task performance. Results suggest that this approach is flexible and offers promise for efficient implementation.